To determine the least and greatest possible measures of angle \( \angle ADC \), let's analyze the situation.
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Understanding the Angles:
- \( \angle ADB \) and \( \angle BDC \) are acute angles that share ray \( DB \).
- Both angles \( \angle ADB \) and \( \angle BDC \) have the same measure, which we will denote as \( x \).
- Since they are acute angles, \( x \) must be less than \( 90^\circ \).
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Relationship of Angles:
- The total measure of angle \( \angle ADC \) is the sum of the two angles: \[ \angle ADC = \angle ADB + \angle BDC = x + x = 2x \]
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Determining Minimum and Maximum Values:
- Since \( x \) is a whole number and less than \( 90^\circ \), the smallest possible value of \( x \) can be \( 1^\circ \) (but note that the angles must be acute, so we are looking for valid options).
- The least possible value of \( x = 1^\circ \) yields: \[ \angle ADC = 2(1) = 2^\circ \]
- Similarly, for the greatest possible value, since \( x \) must be less than \( 90^\circ \) and an integer, the largest possible value of \( x \) can be \( 89^\circ \): \[ \angle ADC = 2(89) = 178^\circ \]
- Since \( x \) is a whole number and less than \( 90^\circ \), the smallest possible value of \( x \) can be \( 1^\circ \) (but note that the angles must be acute, so we are looking for valid options).
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Conclusion: Given that both angles are acute and can only be whole numbers, we confirm the minimum and maximum measures of \( \angle ADC \):
- Least possible measure: \( 2^\circ \)
- Greatest possible measure: \( 178^\circ \)
Thus, the final answer is:
Least: 2°; Greatest: 178°